398 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XVII By use of (1) and (2), theorems I-IV are proved. Theorem VIII is stated, and VII is proved. Employing two diagrams and working to base 2, he showed that 231 — 1 is a prime. Lucas27 considered a product m^p^r". . . of powers of primes, no one dividing Q. SetA = (a-&)2, (A/p)=0, ^l-A^/2 (mod p), Then ^t=0 (mod m) for t—\l/(m). The ranks n of terms un divisible by m are multiples of a certain divisor JJL of \l/(m) . This /i is the exponent to which a or 6 belongs modulo m. The case 6 = 1 gives Euler's generalization of Fermat's theorem. The primality test23 is reproduced and applied to show that 219 — 1 is a prime. Lucas28 considered the series of Pisano. Taking a, 6 = (!=*= A/5) /2, we have Ui=u2 — l, ^3 = 2, etc. According as n is odd or even the divisors of u3n/un are divisors of 5#2 — 3y2 or 5z2+3?/2; those of u±n/u2n are divisors of 5x2 — 2y2 or 5#24-2y2; those of v3n/vn are divisors of x2+3?/2 or x2 — 3t/2; those of v2n are divisors of x*+2y2 or z2 — 2y2; those of w2n/wn are divisors of a?2+52/2 or x* — 5y2. The law V of repetition of primes and theorem III are stated. The law VIII of apparition of primes now takes the following form: If p is a prime lOg^ 1, wp_i is divisible by p ; if p is a prime Wq=*= 3, up+i is divisible by p. The test18 for the primality of A is given and applied to show that 2127 — 1 and 231 — 1 are primes. There is a table of prime factors of un for n^60. Finally, 4upn/un is expressible in the form x2 — py2 or S^+py2 according as the prune p is of the form 4q+l or 4#+3. Lucas29 considered the series defined by rn+1 = rn2 — 2, /1+V5V Let A=3 or 9 (mod 10), g=0 (mod 4); or A=7, 9 (mod 10), q^l( mod 4); or Asl, 7 (mod 10), g=2 (mod 4); or A==l, 3 (mod 10), g==3 (mod 4). Then p = 2qA — 1 is a prime if the rank of the first term divisible by p is 5; if a (a<#) is the rank of the first term divisible by p, the divisors of p are either of the form* 2aAk-\-l, or of the forms of the divisors of x2—2y2 and x2 — 2 Ay2. Corresponding tests are given for 2QA + 1 and 3qA — 1 . The first part of the theorem of Pepin30 for testing the primality of an = 22n+l follows from theorem VII with a = 5, 6 = 1, p = an; the second part follows from the reciprocity theorem and the form of an — 1. For A—p, let the above rl become r. When p^= 7 or 9 (mod 10) and p is a prime, then 2p — I is a prime if and only if r^O (mod 2p — 1). When p = 4q-\~3 is a prime, 2p+l is a prime if and only if 2P=1 (mod 2p + l). When p =4g+3 is a prime, 2p — 1 is a prime if and only if 27Comptcs Rendus Paris, 84, 1877, 439-442. Corrected by Carmichaol.89 28Bull. Bibl. Storia Sc. Mat. e Pis., 10, 1877, 129-170. Reprinted as " Recherches sur plusieurs ouvrages de L6onard de Pise." Cf. von Sterneck21 of Ch. XIX. 29Assoc. fran$. avanc. sc., 6, 1877, 159-166. "Corrected to 2aAK=±l in Lucas39; see Lucas.42 30Comptes Rendus Paris, 85, 1877, 329-331. See Ch. XV, Pepin19, Lucas,18' 22 Proth.23